Distribute a certain number of commodities to a certain number of people according to certain standards. If there is surplus in commodities, it is called surplus; if there is insufficient commodities, it is called deficit. Due to different standards, the results are different, and the number and number of people can be found through the quantitative relationship. This application problem is called profit and loss problem.

Basic characteristics: the total number of objects and the total number of people (or groups) remain unchanged.

There are three basic types of questions: ① one has a remainder and the other has an insufficiency; (2) When there is a remainder twice; (3) when two times are not enough.

Today, let's first understand the first kind of problems: one has a remainder and the other has a deficiency.

Example 1

The third grade teacher gave candy to the children. If each student gives four, he will find three more. If each student gave five, he found that two were missing. How many children are there? How many sweets are there?

Analysis:

Divide candy for the first time. If everyone divides 4, there are still 3 left.

For the second candy distribution, in order for everyone to get five, you need to give everyone 1 (how did you get the 1 here? 5-4= 1)。 But there is not enough candy now, so we have to make up two pieces to ensure that everyone can get 1 piece.

So, if you want to make sure that everyone gets 1 candy, the second time you get candy, you have to get five candy (how did you get five candy here? 3 left in the first time+2 made up in the second time).

How many children does a * * have?

5÷ 1=5 (pieces)

How many sweets are there?

5×4+3=23 or 5×5-2=23.

Example 2

The teacher brought a batch of saplings and distributed them to some students for planting. Each student gets a tree at a time, and each tree is distributed downwards. When there are 12 trees left, it is not enough for each student to share one tree. If you bring eight trees, each student only needs to plant 65,438+00 trees. How many students are going to plant trees? How many primitive saplings are there?

Analysis:

When the saplings are divided for the first time, each person is divided into 9 trees (because the topic says, "If you bring 8 more trees, each student will plant 10 trees"), and the second time, one tree is divided into 10 trees.

The second time, everyone must be assigned to 1 sapling, and 20 saplings must be assigned (how did the 20 saplings here come from? 65438+ 02 trees left for the first time+8 trees added for the second time).

Give students 20 seedlings, each person 1 tree. How many students are there?

20÷ 1=20 (person)

How many seedlings are there?

20×9+ 12= 192 (tree) or 20× 10-8= 192 (tree)

Example 3

The school bought a batch of sporting goods, and the badminton racket is twice as big as the table tennis racket. Distribute to students, each component has 5 pairs of table tennis rackets, table tennis rackets 15 pairs and badminton rackets 14 pairs, with 30 pairs missing. Q: How many pairs of badminton rackets and table tennis rackets did Xueersi School buy?

Analysis:

There are two kinds of sporting goods in the title, and there is a multiple relationship between the two kinds of sporting goods, so we might as well make a conversion (because badminton rackets are twice as big as table tennis rackets, we can replace two pairs of badminton rackets with a pair of table tennis rackets, and the number of badminton rackets in the title can be divided by two to convert into the number relationship of table tennis rackets).

The original problem becomes: each component has 5 sets of table tennis bats, 15 sets of table tennis bats, and each component has 7 sets of table tennis bats, with a difference of 15 sets.

When the table tennis bat is divided for the first time, there are 5 pairs in each group; The second time, if each group is divided into 2 pairs (how did you get the 2 here? 7 pairs in each group for the second time-5 pairs in each group for the first time), then 7 pairs in each group.

Make sure that each group is divided into 2 pairs and 30 pairs (how did you get the 30 pairs here? 15 pairs left in the first time+15 pairs added in the second time).

There are 30 sets of rackets, 2 sets in each group, so how many sets are there?

30÷2= 15 (group)

How many pairs of original table tennis bats are there?

5× 15+ 15=90 (vice)

How many badminton rackets are there?

90×2= 180 (deputy)

Example 4

Xiao Qiang goes to school from home. If he walks 50 meters per minute, he will be three minutes late for class. If you walk 60 meters per minute, you can get to school 2 minutes earlier than the class time. What is the distance from Xiao Qiang's home to school?

Analysis:

As shown in the figure, each small line segment represents 1 minute.

There are two invariable quantities in the problem: ① the distance from home to school remains unchanged; (2) The time from home to the bell remains the same.

For the first time, after the bell rings, it is 50× 3 =150m away from the school;

The second time, I arrived at school 2 minutes before the bell rang, which was equivalent to walking 60×2= 120 meters outside the school.

The difference between starting from home and ringing the bell for class 150+ 120=270 meters.

Now just look at the time from home to the bell ringing (because this time is constant).

Why is the difference 270 meters? Because the second walk is 10 meter per minute. How many minutes has this 270 meters been allocated? 270÷ 10=27 (point)

Therefore, it took 27 minutes to walk from home to school when the bell rang.

The distance from home to school is 50×(27+3)= 1500 (m) or 60×(27-2)= 1500 (m).

Summary profit and loss problem formula

A surplus (surplus), a shortage (deficit), the formula can be used:

(surplus+deficit) ÷ (the difference between two distributions per person) = number of people.